1
00:00:00,521 --> 00:00:01,020
OK.

2
00:00:01,020 --> 00:00:06,580
So that's our introduction
to the subject.

3
00:00:06,580 --> 00:00:08,820
Now, we have to get going.

4
00:00:08,820 --> 00:00:16,860
We have to explore how to set
up this scattering problem.

5
00:00:16,860 --> 00:00:21,990
So equations that
we need to solve.

6
00:00:21,990 --> 00:00:23,640
Well, what are the equations?

7
00:00:23,640 --> 00:00:26,910
We will have a
Hamiltonian, which

8
00:00:26,910 --> 00:00:34,280
is p squared over 2m
plus a potential v of r.

9
00:00:37,200 --> 00:00:41,040
Need not be yet a
central potential.

10
00:00:41,040 --> 00:00:44,850
As usual, we will think
of a wave function that

11
00:00:44,850 --> 00:00:50,060
depends on r and t
that will be written

12
00:00:50,060 --> 00:00:58,410
as a psi that depends on r times
e to the minus iet over h bar.

13
00:00:58,410 --> 00:01:01,500
An energy eigenstate.

14
00:01:01,500 --> 00:01:04,980
And then the equation
that you have to solve,

15
00:01:04,980 --> 00:01:13,570
the time independent the
Schrodinger equation, becomes

16
00:01:13,570 --> 00:01:21,730
minus h squared
over 2m Laplacian

17
00:01:21,730 --> 00:01:34,730
plus v of r acting on psi
of r is equal to e psi of r.

18
00:01:34,730 --> 00:01:39,260
So these are the equations
that you have seen already

19
00:01:39,260 --> 00:01:42,020
endlessly in quantum mechanics.

20
00:01:42,020 --> 00:01:46,220
These are equations we
write to get warmed up,

21
00:01:46,220 --> 00:01:50,910
and we just repeat
for ourselves that we

22
00:01:50,910 --> 00:01:53,310
have a Hamiltonian
in the picture where

23
00:01:53,310 --> 00:01:57,630
the particles are scattering
off of a central potential.

24
00:01:57,630 --> 00:02:00,300
Not a central
potential, in fact,

25
00:02:00,300 --> 00:02:03,240
just often of a potential.

26
00:02:03,240 --> 00:02:05,520
Work with energy
eigenstates, and these

27
00:02:05,520 --> 00:02:08,910
are going to be the equation
for the energy eigenstates.

28
00:02:08,910 --> 00:02:13,140
Now, let's write the first
picture of the scattering

29
00:02:13,140 --> 00:02:17,550
process as some
sort of target here

30
00:02:17,550 --> 00:02:24,810
or potential and
particles that come in

31
00:02:24,810 --> 00:02:28,690
and scatter off
of this potential.

32
00:02:28,690 --> 00:02:34,350
So we're looking for
energy eigenstates.

33
00:02:34,350 --> 00:02:41,400
And we will try to identify
our energy eigenstates,

34
00:02:41,400 --> 00:02:44,760
and what we're going to
assume is that this potential

35
00:02:44,760 --> 00:02:48,895
is finite range as well.

36
00:02:48,895 --> 00:02:49,395
Range.

37
00:02:53,540 --> 00:02:56,150
Finite range we can
deal with potentials

38
00:02:56,150 --> 00:02:59,450
that fall off relatively fast.

39
00:02:59,450 --> 00:03:03,640
Already, the coolant potential
doesn't fall off that fast,

40
00:03:03,640 --> 00:03:06,070
but potentials that fall faster
and the cooler potentials

41
00:03:06,070 --> 00:03:08,840
are potentials that
are just localized,

42
00:03:08,840 --> 00:03:14,430
which is pretty common
if you have an atom

43
00:03:14,430 --> 00:03:16,730
and you scatter
things off of it.

44
00:03:16,730 --> 00:03:20,810
If it's a neutral atom, the
potential due to the atom

45
00:03:20,810 --> 00:03:24,770
is zero outside the atom, but as
soon as you go inside the atom

46
00:03:24,770 --> 00:03:29,270
you start to experience all
the electrical forces that

47
00:03:29,270 --> 00:03:32,060
are due to the nucleus
and the electrons.

48
00:03:32,060 --> 00:03:37,210
So a finite range
potential, and we're

49
00:03:37,210 --> 00:03:40,260
going to think of
solutions that--

50
00:03:40,260 --> 00:03:44,940
OK, away from the finite
range our plane waves,

51
00:03:44,940 --> 00:03:47,980
solutions of constant
energy specified

52
00:03:47,980 --> 00:03:53,960
with perhaps some momentum,
so we will think of e as h

53
00:03:53,960 --> 00:03:57,460
squared k squared over 2m.

54
00:04:01,410 --> 00:04:06,600
This is a way of thinking of
the energy of a given energy

55
00:04:06,600 --> 00:04:07,500
eigenstate.

56
00:04:07,500 --> 00:04:11,830
So it's another
label for the energy.

57
00:04:11,830 --> 00:04:13,920
This one, it looks like
I've done something

58
00:04:13,920 --> 00:04:19,769
but I haven't done much except
to begin an intuition process

59
00:04:19,769 --> 00:04:22,230
in your head in
which somehow these

60
00:04:22,230 --> 00:04:27,240
are going to be related
to energy eigenstates that

61
00:04:27,240 --> 00:04:34,440
have some momentum as they
propagate all over space.

62
00:04:34,440 --> 00:04:38,540
So if I write that, I
could just simply put

63
00:04:38,540 --> 00:04:41,900
this on the left hand
side and get an equation

64
00:04:41,900 --> 00:04:45,200
that is kind of nicer.

65
00:04:45,200 --> 00:04:52,040
Minus h squared
over 2m Laplacian

66
00:04:52,040 --> 00:05:04,080
squared plus k squared plus
v of r psi of r equals zero.

67
00:05:13,490 --> 00:05:25,770
OK, it's not really
a matter scattering

68
00:05:25,770 --> 00:05:30,090
of solving this
equation at this moment.

69
00:05:30,090 --> 00:05:34,740
There is infinitely many
solutions of this equation,

70
00:05:34,740 --> 00:05:38,250
and most of them may
not be relevant for us.

71
00:05:38,250 --> 00:05:41,640
We're not trying to find every
solution of this equation.

72
00:05:41,640 --> 00:05:43,740
We're trying to
find solutions that

73
00:05:43,740 --> 00:05:46,200
have something to
do with physics,

74
00:05:46,200 --> 00:05:49,710
and you've done
that when you had

75
00:05:49,710 --> 00:05:51,760
potentials in one dimension.

76
00:05:51,760 --> 00:05:55,680
And that intuition is
going to prove invaluable.

77
00:05:55,680 --> 00:05:58,440
So when you have a potential in
one dimension, you didn't say,

78
00:05:58,440 --> 00:06:01,170
OK I'm going to find all
the energy eigenstates.

79
00:06:01,170 --> 00:06:04,440
You said, let's
search for things

80
00:06:04,440 --> 00:06:09,150
that are reasonable and
physically motivated,

81
00:06:09,150 --> 00:06:13,380
so you put in a wave
that was moving in

82
00:06:13,380 --> 00:06:17,550
and you said, OK, this
wave is a solution

83
00:06:17,550 --> 00:06:21,630
until it reaches this point
where it just stops being

84
00:06:21,630 --> 00:06:23,850
a solution and you need more.

85
00:06:23,850 --> 00:06:31,710
If you put in this wave, you
will generate a reflected wave

86
00:06:31,710 --> 00:06:34,900
and a transmitted wave.

87
00:06:34,900 --> 00:06:38,770
Those two waves are
going to be generated,

88
00:06:38,770 --> 00:06:42,410
and then you write an
ansatz for this wave,

89
00:06:42,410 --> 00:06:48,750
some coefficient a e to the
ikx, b e to the minus ikx, c

90
00:06:48,750 --> 00:06:51,690
e to the other, and then
you solve your equation.

91
00:06:51,690 --> 00:07:00,690
So we need to do the same thing
with this kind of equation

92
00:07:00,690 --> 00:07:02,280
and this kind of potential.

93
00:07:02,280 --> 00:07:06,540
We have to set up
some sort of situation

94
00:07:06,540 --> 00:07:10,920
where we have the physics
intuition of a wave that

95
00:07:10,920 --> 00:07:16,620
is coming in and then whatever
the system will do to that wave

96
00:07:16,620 --> 00:07:19,365
to upgrade it into
a full solution.

97
00:07:22,220 --> 00:07:27,880
So that's what we want to do
here in analogy to that thing.

98
00:07:27,880 --> 00:07:30,070
This could be called
the incoming wave,

99
00:07:30,070 --> 00:07:33,100
and this whole thing the
reflected and the transmitted

100
00:07:33,100 --> 00:07:36,100
could be called the scattered
wave, the thing that

101
00:07:36,100 --> 00:07:40,430
gets produced by the
scattering process.

102
00:07:40,430 --> 00:07:48,100
So if you had that there is lots
of solutions of that equation,

103
00:07:48,100 --> 00:07:55,720
if v was identical to zero,
if you had no potential,

104
00:07:55,720 --> 00:07:59,680
you could have
lots of solutions,

105
00:07:59,680 --> 00:08:03,880
because in fact if the
potential is not zero,

106
00:08:03,880 --> 00:08:08,950
plane waves are always
solutions without any potential.

107
00:08:08,950 --> 00:08:11,860
Particles that move as
plane waves so if equal

108
00:08:11,860 --> 00:08:25,010
zero, plane waves of the
form psi equal e to the ikx

109
00:08:25,010 --> 00:08:39,779
are solutions with k equal
the square root of k dot k.

110
00:08:45,420 --> 00:08:48,358
Or with k squared
equals k dot k.

111
00:08:56,010 --> 00:08:58,855
Plane waves are always
solutions of this equation.

112
00:09:01,840 --> 00:09:08,440
So if plane waves are solutions
of this equation for v

113
00:09:08,440 --> 00:09:12,960
equal to zero, this
is the same thing

114
00:09:12,960 --> 00:09:18,990
as saying here
that ae to the ikx

115
00:09:18,990 --> 00:09:22,650
is a solution of this
equation as long as you

116
00:09:22,650 --> 00:09:24,060
don't hit the potential.

117
00:09:27,130 --> 00:09:30,310
So here, we're going to do
something quite similar.

118
00:09:30,310 --> 00:09:35,070
We're going to say that we're
going to put in an incident

119
00:09:35,070 --> 00:09:48,320
wave function, and I will
instead of writing psi,

120
00:09:48,320 --> 00:09:51,760
I will call it 5x.

121
00:09:51,760 --> 00:09:57,895
The incident wave function 5x is
going to be just e to the ikz.

122
00:10:04,390 --> 00:10:05,620
So it's a wave function.

123
00:10:05,620 --> 00:10:09,500
I call it phi to
distinguish it from psi.

124
00:10:09,500 --> 00:10:13,180
Psi in general is
a full solution

125
00:10:13,180 --> 00:10:16,550
of the Schrodinger equation.

126
00:10:16,550 --> 00:10:19,450
That's our understanding of psi.

127
00:10:19,450 --> 00:10:24,020
So phi, it reminds you that
well, it's some wave function.

128
00:10:24,020 --> 00:10:25,740
I'm not sure it's a solution.

129
00:10:25,740 --> 00:10:29,710
In fact, it probably is
not the solution as soon

130
00:10:29,710 --> 00:10:34,540
as you have the potential
different from zero.

131
00:10:34,540 --> 00:10:38,110
So is this common, we
forget that ksv equals zero,

132
00:10:38,110 --> 00:10:43,450
and now we put an incident wave
function which is of this form.

133
00:10:43,450 --> 00:10:48,790
This solves the equation
as long as you're away

134
00:10:48,790 --> 00:10:50,710
from the potential.

135
00:10:50,710 --> 00:11:00,460
This is true, it's a solution
of the Schrodinger equation

136
00:11:00,460 --> 00:11:07,610
away from v of r.

137
00:11:07,610 --> 00:11:11,960
Away from v of r means
wherever v of r is equal zero,

138
00:11:11,960 --> 00:11:14,900
you have a solution.

139
00:11:14,900 --> 00:11:19,460
Nevertheless, so if we call
the range of the potential--

140
00:11:19,460 --> 00:11:22,340
let's call the range of the
potential finite range a--

141
00:11:25,320 --> 00:11:29,430
that is to take that if
there is an origin here

142
00:11:29,430 --> 00:11:34,530
up to a radius a, there is some
potential and beyond the radius

143
00:11:34,530 --> 00:11:37,930
a the potential vanishes.

144
00:11:37,930 --> 00:11:42,550
So this definitely works.

145
00:11:42,550 --> 00:11:54,960
It's all ks away from vr or
as long as r is greater than a

146
00:11:54,960 --> 00:11:57,540
for whatever value
of z you take.

147
00:11:57,540 --> 00:12:05,650
This is fine, so
here is our wave,

148
00:12:05,650 --> 00:12:09,380
and now this is just
the incident wave.

149
00:12:09,380 --> 00:12:12,560
This is not going to
be the whole solution.

150
00:12:12,560 --> 00:12:15,860
Just like in the one dimensional
case, there must be more.

151
00:12:15,860 --> 00:12:17,630
So what is there more?

152
00:12:17,630 --> 00:12:21,670
And our challenge to
begin with this problem

153
00:12:21,670 --> 00:12:26,700
is to set up what
else could there be.

154
00:12:26,700 --> 00:12:32,390
So looking at it,
you'd say, all right.

155
00:12:32,390 --> 00:12:34,700
So the thing comes in.

156
00:12:34,700 --> 00:12:37,380
If there is
scattering, particles

157
00:12:37,380 --> 00:12:41,130
are sometimes going to go
off in various directions.

158
00:12:41,130 --> 00:12:46,760
So the outgoing wave, here
there were outgoing waves

159
00:12:46,760 --> 00:12:48,890
reflected and transmitted.

160
00:12:48,890 --> 00:12:53,000
The outgoing wave in the three
dimensional scattering problem

161
00:12:53,000 --> 00:12:58,910
should be some sort of spherical
wave moving away from r

162
00:12:58,910 --> 00:13:02,420
equals zero, which
is the origin.

163
00:13:02,420 --> 00:13:05,540
That should be the other
wave that I would write.

164
00:13:05,540 --> 00:13:10,260
So my ansatz should
be that there

165
00:13:10,260 --> 00:13:16,510
is some sort of spherical
wave that is moving away.

166
00:13:16,510 --> 00:13:29,750
So to complete this with
a spherical outgoing wave.

167
00:13:33,590 --> 00:13:40,910
So while here this is
a plane wave moving

168
00:13:40,910 --> 00:13:43,880
in the direction
of the vector k,

169
00:13:43,880 --> 00:13:50,420
if I want to write the spherical
wave, I would write e to the i

170
00:13:50,420 --> 00:13:51,050
just kr.

171
00:13:54,490 --> 00:13:59,230
E to the ikr is
spherically symmetric,

172
00:13:59,230 --> 00:14:02,270
and it propagates radiantly out.

173
00:14:02,270 --> 00:14:04,660
If you remember
as usual that you

174
00:14:04,660 --> 00:14:12,220
have e to the minus iet over h
bar, so you have kr minus et,

175
00:14:12,220 --> 00:14:16,720
that is a wave that
propagates radially out.

176
00:14:16,720 --> 00:14:24,150
So maybe this is kind
of the scattered wave.

177
00:14:24,150 --> 00:14:29,190
This e to the ikr
moving out everywhere

178
00:14:29,190 --> 00:14:30,735
would be your scattered wave.

179
00:14:35,390 --> 00:14:39,830
If that is the
scattered wave, remember

180
00:14:39,830 --> 00:14:44,250
the scattered wave should
solve the Schrodinger equation.

181
00:14:44,250 --> 00:14:46,280
In fact, the sum
of these two should

182
00:14:46,280 --> 00:14:48,450
solve the scattering equation.

183
00:14:48,450 --> 00:14:50,390
On the other hand,
we've seen that this

184
00:14:50,390 --> 00:14:55,370
solves it as long as you're
away from the potential,

185
00:14:55,370 --> 00:14:58,100
and therefore this
should also solve it

186
00:14:58,100 --> 00:15:00,290
if you're away
from the potential.

187
00:15:03,450 --> 00:15:07,340
So I ask you, do you think this
solves the Schrodinger equation

188
00:15:07,340 --> 00:15:09,170
when you're away
from the potential?

189
00:15:14,140 --> 00:15:17,840
Would e to the ikr
solve the equation?

190
00:15:17,840 --> 00:15:21,240
Well for that, you would
need that if you're away

191
00:15:21,240 --> 00:15:26,250
from the potential, do
you have Laplacian of e

192
00:15:26,250 --> 00:15:30,680
to the ikr roughly equals zero?

193
00:15:34,540 --> 00:15:35,190
Is that true?

194
00:15:44,550 --> 00:15:46,850
Would it solve it.

195
00:15:46,850 --> 00:15:47,520
No.

196
00:15:47,520 --> 00:15:48,630
It doesn't solve it.

197
00:15:48,630 --> 00:15:52,620
Doesn't even come
close to solving it.

198
00:15:52,620 --> 00:15:57,990
It's pretty bad, but the reason
it's bad is physically clear.

199
00:15:57,990 --> 00:16:04,890
This wave as it expands out
must become weaker and weaker

200
00:16:04,890 --> 00:16:09,720
so that the probability
flux remains constant.

201
00:16:09,720 --> 00:16:11,910
You know, you don't
want an accumulation

202
00:16:11,910 --> 00:16:16,110
of probability between a shell
at one kilometer and a shell

203
00:16:16,110 --> 00:16:17,230
at two kilometers.

204
00:16:17,230 --> 00:16:22,190
So whatever flux is going out
from the shell in one kilometer

205
00:16:22,190 --> 00:16:24,930
should be going out
of the bigger shell.

206
00:16:24,930 --> 00:16:29,280
So therefore, it
should fall off with r.

207
00:16:29,280 --> 00:16:30,300
Oh, I'm sorry.

208
00:16:30,300 --> 00:16:34,320
I didn't write this well.

209
00:16:34,320 --> 00:16:39,090
So if you are going
to have this to be

210
00:16:39,090 --> 00:16:43,320
a solution of the Schrodinger
equation outside of vr

211
00:16:43,320 --> 00:16:49,250
equals zero, you should
have Laplacian k squared

212
00:16:49,250 --> 00:16:51,570
of this thing equal to zero.

213
00:16:51,570 --> 00:16:55,210
So if this is equal to zero and
the potential is equal to zero,

214
00:16:55,210 --> 00:16:57,480
the whole thing
is equal to zero.

215
00:16:57,480 --> 00:17:00,030
So we need this to hold.

216
00:17:00,030 --> 00:17:03,880
But even this one, of
course, is not true.

217
00:17:03,880 --> 00:17:06,010
It is just absolutely not true.

218
00:17:06,010 --> 00:17:09,599
The one that works
is the following.

219
00:17:12,690 --> 00:17:23,329
Laplacian plus k squared
of e to the ikr over r

220
00:17:23,329 --> 00:17:28,830
is equal to zero for
r different from zero.

221
00:17:28,830 --> 00:17:32,750
This is a computation
I think you guys have

222
00:17:32,750 --> 00:17:35,390
done before when
you were studying

223
00:17:35,390 --> 00:17:37,880
the Hermiticity of p squared.

224
00:17:37,880 --> 00:17:40,310
You ended up doing
this kind of things.

225
00:17:40,310 --> 00:17:45,860
This Laplacian produces a delta
function at r equals zero,

226
00:17:45,860 --> 00:17:49,620
but r equals zero is not the
place we're interested in.

227
00:17:49,620 --> 00:17:53,780
We're trying to find
how the waves look away

228
00:17:53,780 --> 00:17:57,420
from the scattering center.

229
00:17:57,420 --> 00:18:02,240
So we need this to
hold away from r

230
00:18:02,240 --> 00:18:06,860
equals zero, in fact,
bigger for r bigger than a.

231
00:18:06,860 --> 00:18:11,630
One way of checking
this kind of thing

232
00:18:11,630 --> 00:18:18,200
is to remember that the
Laplacian of a function of r

233
00:18:18,200 --> 00:18:27,290
is in fact one over r d
second dr squared r times f.

234
00:18:27,290 --> 00:18:32,330
That's a neat formula for the
Laplacian of a function that

235
00:18:32,330 --> 00:18:33,620
just depends on r.

236
00:18:33,620 --> 00:18:36,960
If it depends on theta and
phi, it's more complicated.

237
00:18:36,960 --> 00:18:40,730
But with this function, it
becomes a one line calculation

238
00:18:40,730 --> 00:18:44,450
to do this, and the r
here is just fantastic,

239
00:18:44,450 --> 00:18:48,830
because by the time you
multiply by r this function is

240
00:18:48,830 --> 00:18:49,930
just exponential.

241
00:18:49,930 --> 00:18:55,820
You take two derivatives,
you get minus k squared,

242
00:18:55,820 --> 00:18:58,940
and then the r gets
canceled as well

243
00:18:58,940 --> 00:19:02,490
and everything works
beautifully here.

244
00:19:02,490 --> 00:19:09,770
And so this equation holds OK.

245
00:19:09,770 --> 00:19:15,890
And then we do have a
possible scattered wave.

246
00:19:15,890 --> 00:19:18,410
So we're almost there.

247
00:19:18,410 --> 00:19:29,550
We can write the scattering
wave size scattering

248
00:19:29,550 --> 00:19:45,470
of x could be e to the ikr over
r and then leave it at that.

249
00:19:45,470 --> 00:19:48,120
But this would not
be general enough.

250
00:19:48,120 --> 00:19:53,470
There is no reason why
this wave would not

251
00:19:53,470 --> 00:19:56,270
depend also on theta and phi.

252
00:19:59,560 --> 00:20:03,130
Here is the z
direction, and there's

253
00:20:03,130 --> 00:20:07,450
points with angle theta, and if
you rotate it with some angle

254
00:20:07,450 --> 00:20:13,930
phi here, and therefore
this function could as well

255
00:20:13,930 --> 00:20:16,420
depend on theta and phi.

256
00:20:16,420 --> 00:20:24,590
So we'll include that
factor f of theta and phi.

257
00:20:31,550 --> 00:20:34,000
Now you would say,
look, you have

258
00:20:34,000 --> 00:20:37,090
a nice solution of the
Schrodinger equation

259
00:20:37,090 --> 00:20:45,110
already here, and what
should you be doing?

260
00:20:45,110 --> 00:20:48,020
Why do you add this factor?

261
00:20:48,020 --> 00:20:52,100
With this factor, it may
not be anymore a solution

262
00:20:52,100 --> 00:20:53,537
of the Schrodinger equation.

263
00:20:53,537 --> 00:20:55,745
The Schrodinger equation is
going to have a Laplacian

264
00:20:55,745 --> 00:20:59,040
and it's going to
be more complicated.

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00:20:59,040 --> 00:21:03,140
Well, this is true and we
will see that better soon,

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00:21:03,140 --> 00:21:07,040
but this will remain
an approximate solution

267
00:21:07,040 --> 00:21:14,690
for r much bigger than a is the
leading term of the solution,

268
00:21:14,690 --> 00:21:19,385
leaving term of the solution.

269
00:21:30,850 --> 00:21:36,660
So if you have a leading
term of a solution here,

270
00:21:36,660 --> 00:21:38,810
this is all you want.

271
00:21:38,810 --> 00:21:44,270
You are working at r much bigger
than a, and your whole wave

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00:21:44,270 --> 00:21:46,880
and finally be written.

273
00:21:46,880 --> 00:21:53,100
So your full wave psi, or your
full energy eigenstates psi

274
00:21:53,100 --> 00:21:57,380
of rt, is going to be
equal, approximately

275
00:21:57,380 --> 00:22:00,740
equal to e to the ik--

276
00:22:00,740 --> 00:22:03,260
well, I'll write it this way.

277
00:22:03,260 --> 00:22:09,680
Phi of r, that incident
wave we wrote--

278
00:22:09,680 --> 00:22:12,020
I wrote if x there, I'm sorry.

279
00:22:12,020 --> 00:22:17,940
Plus psi scattering of--

280
00:22:17,940 --> 00:22:22,320
I should decide x or r.

281
00:22:22,320 --> 00:22:26,550
Let's call this r.

282
00:22:26,550 --> 00:22:27,870
And I should call this r.

283
00:22:33,826 --> 00:22:37,930
Psi of r plus psi
scattering of r,

284
00:22:37,930 --> 00:22:47,460
and it's therefore equal to e
to the ikz plus f of theta phi e

285
00:22:47,460 --> 00:22:56,160
to the ikr over r, and this is
only valid for r much bigger

286
00:22:56,160 --> 00:22:57,690
than a.

287
00:22:57,690 --> 00:22:59,520
Far away.

288
00:22:59,520 --> 00:23:05,430
This had an analog in our
problem in one dimension.

289
00:23:05,430 --> 00:23:12,860
In one dimension, you set up a
wave and you put here the wave

290
00:23:12,860 --> 00:23:17,430
and there is a reflected wave,
and there's a transmitted wave,

291
00:23:17,430 --> 00:23:21,180
and in setting the problem, say
this is valid far to the left,

292
00:23:21,180 --> 00:23:25,530
far to the left meaning at least
to the left of the barrier,

293
00:23:25,530 --> 00:23:28,110
this is valid to the right.

294
00:23:28,110 --> 00:23:33,300
And these were exact
solutions in this region here.

295
00:23:33,300 --> 00:23:37,470
You can't do that well, but
you can do reasonably well,

296
00:23:37,470 --> 00:23:44,850
you can write solutions that
are leading term accurate

297
00:23:44,850 --> 00:23:48,370
as long as you are far away.

298
00:23:48,370 --> 00:23:54,630
So this is the first step in
this whole process in which we

299
00:23:54,630 --> 00:23:59,830
are setting up the
wave functions that

300
00:23:59,830 --> 00:24:01,840
is the most important equation.

301
00:24:01,840 --> 00:24:05,810
This is the way we're going
to try to find solutions.

302
00:24:09,030 --> 00:24:11,640
Now, when you try
to find solutions

303
00:24:11,640 --> 00:24:13,770
at the end of the
day, you will have

304
00:24:13,770 --> 00:24:19,050
to work in the region
r going to zero.

305
00:24:19,050 --> 00:24:22,410
So sooner or later
we'll get there.

306
00:24:22,410 --> 00:24:25,980
But for the time being, we
have all the information

307
00:24:25,980 --> 00:24:30,510
about what's going on far
away, and from there we

308
00:24:30,510 --> 00:24:33,360
can get most of what we need.

309
00:24:33,360 --> 00:24:38,160
In particular, this
f of theta and phi

310
00:24:38,160 --> 00:24:42,510
is the quantity we
need to figure out.

311
00:24:42,510 --> 00:24:47,160
This f of theta
and phi is called

312
00:24:47,160 --> 00:24:49,770
the scattering amplitude.

313
00:24:49,770 --> 00:24:58,040
F of theta and phi called
scattering amplitude.